I found a period mentioning of the special GMGA helmet (i.e., the 'scroll' helmet) which includes a reference of the order for its issue. Actually I wanted to research the Würt. 4. mounted carriage company but found that I had ordered the wrong book from the "Württembergs Heer im Weltkrieg" series. So I used the time and read through the parts that I had ordered and found some nice background information over the formation of the GMGAs.
The mentioning of the helmets is following:
"The mountain M.G. personnel received the gray-green uniform of the machinegun units with the unit numbers on the shoulder flaps, and experimentally a lighweight helmet of felt with matte fittings, ventilation and a neck flap (Pr. Kr. Min. Nr. 1960/11. 15 B. 3 D, footnote)."
In the following I will translate what is written about the GMGA formation until 1917, leaving out some lenghty passages (marked ...) and marking my own notes with square brackets. The orders are usually referenced via footnotes:
"Very soon after war begin the need for special machinegun units besides the normal MG issues for the troops fighting in mountain terrain became obvious in the Vosges and the east front. Only whith these it was possible to move firepower quickly to the required places, to support own actions or defend stretched frontlines against superior enemy forces. Upon an urgend request of the OHL [Oberste Heeresleitung, i.e. highest army command], fourty GMGAs numbered 211 to 250 were quickly established by the MG training unit Döberitz beginning in August 1915. 18 of these units had to be deployable by Sept. 9th, the other 22 by Sept. 20th 1915 (Pr. Kr. Min. Nr. 2002/8. 15 A. 2 D from August 21st, 1915). The mobilization of these troop was so important that all other requests had to stand back and the outfitting of other MG units that were planned for August and September was delayed. All personnel and carriage animals that had been sent to Döberitz for other reasons were assigned to the first 18 GMGA units (Pr. Kr. Min. Nr. 976/8. 15 A. 2 D from August 12th, 1915). The following 22 GMGA units were equipped with personnel and horses by the Stellv. Gen. Kos. [assistant general commands?] by Sept. 7th 1915, with Württemberg providing one of these units. The personnel had to be healthy and suitable for action in mountain terrain. The special mountain equipment and the outfit were provided by the Pr. Kr. Min. (Prussian war ministery), guns, harnesses and ammunition by the Gew. Prüf. Kom. [Gewehr Prüf-Kommission?] resp. the field maintenance depot. The mountain MG personnel received the gray-green uniform of the machinegun units with the units numbers on the shoulder flaps, and experimentally a lighweight helmet of felt with matte fittings, ventilation and a neck flap (Pr. Kr. Min. Nr. 1960/11. 15 B. 3 D). The GMGA units were organized in 2 platoons with 3 machineguns each and had a required strength of 4 officers, 175 other ranks, 85 horses (including 48 carriage animals) and 7 vehicles.
Upon this order the personnel for the Württemberg GMGA 250 was assembled at the 1. Ersatz Machinegun company 121 in Münsingen and send to the MG training in Döberitz on Sept. 5th. By Sept. 15th the unit was regarded as mobile and already 14 days later on Oct. 1st it was deployed to the 11. army in order to move to the Oberkommando Mackensen from southern Hungary and participate in the offensive against Serbia.
This way fourty GMGA units were mobilized in a single month. ... Upon orders of the OHL, the GMGA units 235-238, 241 and 243-246 were reorganized to MG companies in March 1916 and attached to infantry regiments as first or second MG companies (Pr. Kr. Min. Nr. 1656/2. 16 A. 2 D). At the same time, the remaining GMGAs were strenghtened by 21 men, 20 horses and 3 double-sided vehicles [wagons with two horses] each, the latter being light Gebirgswagen 15 [evidently an official denotation] or vehicles of the region. ... While this increase of strength became effective, orders of the ADK 11 (army group Mackensen) attached the GMGA units 210, 240 and 247-250 to the 5th Bulgarian division were they served at a quite streched frontline. This deployment required to increase the number of machineguns from 6 to 9 by the end of July 1916. This state remained until early 1917. On May 7th 1917 a verification of the strength of all GMGAs was again required and confirmed at 9 guns per unit. At the same time, a depot with the name "Feldrekrutendepot für G.-M.G.-Abtlgen der Heeresgruppe Scholtz" was established in Alexinac for the GMGA units in Serbia (*).
* Footnote:
Consisting of command, training personnel, marksmen unit, carriage animal unit, urgical unit. Total strength 16 officers, 827 men, 127 horses (including 90 carriage animals) and 7 vehicles. ...
After this reorganization the required strength of a GMGA was 9 machineguns, 4 officers, 247 other ranks, 139 horses (including 80 carriage animals) and 21 vehicles, allthough this strength was never reached due to the increasing shortage of personnel and equipment. "
...
Some conclusions:
* The reason that often only the GMGA units 211-250 are mentioned may be that these were the first batch that was mobilized. Evidently, the lower numbers 201-109 followed later.
* The uniforms were provided by Prussia, which is probably the reason why GMGAs from different states have identical outfit (including helmets).
* The special helmet model was intentional and not (or at least not exclusively) the result of the shortage.
* The GMGAs have a quite large number of horses and carriage animals (139/80 after beefing up in 1917) considering their size similar to companies. In other parts of the document it is described how difficult it was to equip the mountain units with suitable carriage animals. Carriage ponys were even imported from Sweden. So evidently the mobility (and independence from other transport units) was regarded a main asset of a GMGA.
The mentioning of the helmets is following:
"The mountain M.G. personnel received the gray-green uniform of the machinegun units with the unit numbers on the shoulder flaps, and experimentally a lighweight helmet of felt with matte fittings, ventilation and a neck flap (Pr. Kr. Min. Nr. 1960/11. 15 B. 3 D, footnote)."

In the following I will translate what is written about the GMGA formation until 1917, leaving out some lenghty passages (marked ...) and marking my own notes with square brackets. The orders are usually referenced via footnotes:
"Very soon after war begin the need for special machinegun units besides the normal MG issues for the troops fighting in mountain terrain became obvious in the Vosges and the east front. Only whith these it was possible to move firepower quickly to the required places, to support own actions or defend stretched frontlines against superior enemy forces. Upon an urgend request of the OHL [Oberste Heeresleitung, i.e. highest army command], fourty GMGAs numbered 211 to 250 were quickly established by the MG training unit Döberitz beginning in August 1915. 18 of these units had to be deployable by Sept. 9th, the other 22 by Sept. 20th 1915 (Pr. Kr. Min. Nr. 2002/8. 15 A. 2 D from August 21st, 1915). The mobilization of these troop was so important that all other requests had to stand back and the outfitting of other MG units that were planned for August and September was delayed. All personnel and carriage animals that had been sent to Döberitz for other reasons were assigned to the first 18 GMGA units (Pr. Kr. Min. Nr. 976/8. 15 A. 2 D from August 12th, 1915). The following 22 GMGA units were equipped with personnel and horses by the Stellv. Gen. Kos. [assistant general commands?] by Sept. 7th 1915, with Württemberg providing one of these units. The personnel had to be healthy and suitable for action in mountain terrain. The special mountain equipment and the outfit were provided by the Pr. Kr. Min. (Prussian war ministery), guns, harnesses and ammunition by the Gew. Prüf. Kom. [Gewehr Prüf-Kommission?] resp. the field maintenance depot. The mountain MG personnel received the gray-green uniform of the machinegun units with the units numbers on the shoulder flaps, and experimentally a lighweight helmet of felt with matte fittings, ventilation and a neck flap (Pr. Kr. Min. Nr. 1960/11. 15 B. 3 D). The GMGA units were organized in 2 platoons with 3 machineguns each and had a required strength of 4 officers, 175 other ranks, 85 horses (including 48 carriage animals) and 7 vehicles.
Upon this order the personnel for the Württemberg GMGA 250 was assembled at the 1. Ersatz Machinegun company 121 in Münsingen and send to the MG training in Döberitz on Sept. 5th. By Sept. 15th the unit was regarded as mobile and already 14 days later on Oct. 1st it was deployed to the 11. army in order to move to the Oberkommando Mackensen from southern Hungary and participate in the offensive against Serbia.
This way fourty GMGA units were mobilized in a single month. ... Upon orders of the OHL, the GMGA units 235-238, 241 and 243-246 were reorganized to MG companies in March 1916 and attached to infantry regiments as first or second MG companies (Pr. Kr. Min. Nr. 1656/2. 16 A. 2 D). At the same time, the remaining GMGAs were strenghtened by 21 men, 20 horses and 3 double-sided vehicles [wagons with two horses] each, the latter being light Gebirgswagen 15 [evidently an official denotation] or vehicles of the region. ... While this increase of strength became effective, orders of the ADK 11 (army group Mackensen) attached the GMGA units 210, 240 and 247-250 to the 5th Bulgarian division were they served at a quite streched frontline. This deployment required to increase the number of machineguns from 6 to 9 by the end of July 1916. This state remained until early 1917. On May 7th 1917 a verification of the strength of all GMGAs was again required and confirmed at 9 guns per unit. At the same time, a depot with the name "Feldrekrutendepot für G.-M.G.-Abtlgen der Heeresgruppe Scholtz" was established in Alexinac for the GMGA units in Serbia (*).
* Footnote:
Consisting of command, training personnel, marksmen unit, carriage animal unit, urgical unit. Total strength 16 officers, 827 men, 127 horses (including 90 carriage animals) and 7 vehicles. ...
After this reorganization the required strength of a GMGA was 9 machineguns, 4 officers, 247 other ranks, 139 horses (including 80 carriage animals) and 21 vehicles, allthough this strength was never reached due to the increasing shortage of personnel and equipment. "
...
Some conclusions:
* The reason that often only the GMGA units 211-250 are mentioned may be that these were the first batch that was mobilized. Evidently, the lower numbers 201-109 followed later.
* The uniforms were provided by Prussia, which is probably the reason why GMGAs from different states have identical outfit (including helmets).
* The special helmet model was intentional and not (or at least not exclusively) the result of the shortage.
* The GMGAs have a quite large number of horses and carriage animals (139/80 after beefing up in 1917) considering their size similar to companies. In other parts of the document it is described how difficult it was to equip the mountain units with suitable carriage animals. Carriage ponys were even imported from Sweden. So evidently the mobility (and independence from other transport units) was regarded a main asset of a GMGA.